Is there a well-known name for the triangular "half" of the (square) grid graph (see picture below)?
This is actually a special case of my full question: I am studying the crystal base for the $U_q(\mathfrak{sl}_n)$-module with highest weight $(d)=(d,0,\ldots,0)$. The picture above is the case when $n=3$ and $d=4$, where each horizontal (resp. vertical) edge corresponds to the simple root $\alpha_1$ (resp. $\alpha_2$). For my purposes, we can ignore the direction of the edges.
We can visualize these crystals as subsets of the integer lattice in $(n-1)$-dimensions, with each coordinate direction corresponding to a simple root. For my special case (highest weight $(d)$), these graphs seem relatively "nice." Is there a well-known description/construction in graph theoretic terms? (Even in the 2-dimensional case as pictured?) I ask because ultimately I would like to compute the Wiener index of these crystals, and I'm wondering if this is even a tractable problem.